Right Triangles - ACT Math
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In a right triangle ABC, the measure of angle C is greater than 60 degrees. Which of the following statements could describe the measures of angles A and B?
In a right triangle ABC, the measure of angle C is greater than 60 degrees. Which of the following statements could describe the measures of angles A and B?
Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.
Example:
If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.
Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.
Example:
If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.
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A 17 ft ladder is propped against a 15 ft wall. What is the degree measurement between the ladder and the ground?
A 17 ft ladder is propped against a 15 ft wall. What is the degree measurement between the ladder and the ground?
Since all the answer choices are in trigonometric form, we know we must not necessarily solve for the exact value (although we can do that and calculate each choice to see if it matches). The first step is to determine the length of the ground between the bottom of the ladder and the wall via the Pythagorean Theorem: "x2 + 152 = 172"; x = 8. Using trigonometric definitions, we know that "opposite/adjecent = tan(theta)"; since we have both values of the sides (opp = 15 and adj = 8), we can plug into the tangential form tan(theta) = 15/8. However, since we are solving for theta, we must take the inverse tangent of the left side, "tan-1". Thus, our final answer is
Since all the answer choices are in trigonometric form, we know we must not necessarily solve for the exact value (although we can do that and calculate each choice to see if it matches). The first step is to determine the length of the ground between the bottom of the ladder and the wall via the Pythagorean Theorem: "x2 + 152 = 172"; x = 8. Using trigonometric definitions, we know that "opposite/adjecent = tan(theta)"; since we have both values of the sides (opp = 15 and adj = 8), we can plug into the tangential form tan(theta) = 15/8. However, since we are solving for theta, we must take the inverse tangent of the left side, "tan-1". Thus, our final answer is
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What is the sine of the angle between the base and the hypotenuse of a right triangle with a base of 4 and a height of 3?
What is the sine of the angle between the base and the hypotenuse of a right triangle with a base of 4 and a height of 3?
By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.
By rule, this is a 3-4-5 right triangle. Sine = (the opposite leg)/(the hypotenuse). This gives us 3/5.
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The measure of 3 angles in a triangle are in a 1:2:3 ratio. What is the measure of the middle angle?
The measure of 3 angles in a triangle are in a 1:2:3 ratio. What is the measure of the middle angle?
The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.
The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.
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Right triangle 
has an acute angle measuring 
. What is the measure of the other acute angle?
Right triangle
The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle must be 
. We know that a right triangle has one angle equal to 
, and we are told one of the acute angles is 
.
The rest is simple subtraction:
Thus, our missing angle is 
.
The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle must be
The rest is simple subtraction:
Thus, our missing angle is
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Right triangle 
has angles with a ratio of with a ratio of 
. What is the smallest angle in the triangle?
Right triangle
Solving this problem quickly requires that we recognize how to break apart our ratio.
The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle is 
. Additionally, the Right Triangle Acute Angle Theorem states that the two non-right angles in a right triangle are acute; that is to say, the right angle is always the largest angle in a right triangle.
Since this is true, we can assume that 
is represented by the largest number in the ratio of angles. Now consider that the other two angles must also sum to 
. We know therefore that the sum of their ratios must be divisible by 
as well.
Thus, 
.
To find the value of one angle of the ratio, simply assign fractional value to the sum of the ratios and multiply by 
.
, so:
Thus, the shortest angle (the one represented by 
in our ratio of angles) is 
.
Solving this problem quickly requires that we recognize how to break apart our ratio.
The Triangle Angle Sum Theorem states that the sum of all interior angles in a triangle is
Since this is true, we can assume that
Thus,
To find the value of one angle of the ratio, simply assign fractional value to the sum of the ratios and multiply by
Thus, the shortest angle (the one represented by
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is similar to 
. What is the measurement of segment 
?
The word similar means, comparable in measurement, but not equal. The best sides way to compare these two triangles is by looking at the diagonal side of the triangle since it cannot be mistaken for any of the other sides of either triangle. If the larger triangle has a measurement of 7 and the smaller triangle has a measurement of 3.5 for their diagonal sides, then that means the ratio of the larger triangle to the smaller one is 
.
This means that segment EF must be similar to segment AC (look at the orientation). So, since segment DE is similar to segment CB, divide 5 by two to get your answer.
The word similar means, comparable in measurement, but not equal. The best sides way to compare these two triangles is by looking at the diagonal side of the triangle since it cannot be mistaken for any of the other sides of either triangle. If the larger triangle has a measurement of 7 and the smaller triangle has a measurement of 3.5 for their diagonal sides, then that means the ratio of the larger triangle to the smaller one is
This means that segment EF must be similar to segment AC (look at the orientation). So, since segment DE is similar to segment CB, divide 5 by two to get your answer.
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There are two right triangles, Triangle A and Triangle B. Triangle A has sides with lengths of 3, 4, and 5 and Triangle B has only two given side lengths: 3 and 4. The length of the hypotenuse is not given. Are these two triangles congruent?
There are two right triangles, Triangle A and Triangle B. Triangle A has sides with lengths of 3, 4, and 5 and Triangle B has only two given side lengths: 3 and 4. The length of the hypotenuse is not given. Are these two triangles congruent?
1. Find the length of the missing hypotenuse:
You can do this one of two ways:
-
Using the special 3-4-5 right triangle, you can infer that the missing hypotenuse is 5.
-
By using the Pythagorean Theorem, you can solve for the length of the hypotenuse:
In this case:
2. Using the meaning of congruent (the exact same three angles and sides), determine if these two triangles fit this meaning:
They do, because both are 3-4-5 right triangles, and thus must have corresponding and equal angles due to trigonometric properties.
1. Find the length of the missing hypotenuse:
You can do this one of two ways:
-
Using the special 3-4-5 right triangle, you can infer that the missing hypotenuse is 5.
-
By using the Pythagorean Theorem, you can solve for the length of the hypotenuse:
In this case:
2. Using the meaning of congruent (the exact same three angles and sides), determine if these two triangles fit this meaning:
They do, because both are 3-4-5 right triangles, and thus must have corresponding and equal angles due to trigonometric properties.
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There are two similar right triangles. One has side lengths of 6, 8, and 10. The other has side lengths of 24, and 18. What is the length of the missing side?
There are two similar right triangles. One has side lengths of 6, 8, and 10. The other has side lengths of 24, and 18. What is the length of the missing side?
1. Since the two triangles are similar, find the ratio of the two triangles to each other:
In this case, both triangles are multiples of a 
special right triangle.
and
The ratio of the triangles is 
2. Use the ratio you found to solve for 
, or the length of the missing side:
1. Since the two triangles are similar, find the ratio of the two triangles to each other:
In this case, both triangles are multiples of a
and
The ratio of the triangles is
2. Use the ratio you found to solve for
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You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
You have two right triangles that are similar. The base of the first is 6 and the height is 9. If the base of the second triangle is 20, what is the height of the second triangle?
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
Similar triangles are proportional.
Base1 / Height1 = Base2 / Height2
6 / 9 = 20 / Height2
Cross multiply and solve for Height2
6 / 9 = 20 / Height2
6 * Height2= 20 * 9
Height2= 30
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A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
A right triangle is defined by the points (1, 1), (1, 5), and (4, 1). The triangle's sides are enlarged by a factor of 3 to form a new triangle. What is the area of the new triangle?
The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
The points define a 3-4-5 right triangle. Its area is A = 1/2bh = ½(3)(4) = 6. The scale factor (SF) of the new triangle is 3. The area of the new triangle is given by Anew = (SF)2 x (Aold) =
32 x 6 = 9 x 6 = 54 square units (since the units are not given in the original problem).
NOTE: For a volume problem: Vnew = (SF)3 x (Vold).
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On a flat street, a light pole 36 feet tall casts a shadow that is 9 feet long. At the same time of day, a nearby light pole casts a shadow that is 6 feet long. How many feet tall is the second light pole?
On a flat street, a light pole 36 feet tall casts a shadow that is 9 feet long. At the same time of day, a nearby light pole casts a shadow that is 6 feet long. How many feet tall is the second light pole?
Start by drawing out the light poles and their shadows.
In this case, we end up with two similar triangles. We know that these are similar triangles because the question tells us that these poles are on a flat surface, meaning angle B and angle E are both right angles. Then, because the question states that the shadow cast by both poles are at the same time of day, we know that angles C and F are equivalent. As a result, angles A and D must also be equivalent.
Since these are similar triangles, we can set up proportions for the corresponding sides.
Now, solve for 
by cross-multiplying.
Start by drawing out the light poles and their shadows.
In this case, we end up with two similar triangles. We know that these are similar triangles because the question tells us that these poles are on a flat surface, meaning angle B and angle E are both right angles. Then, because the question states that the shadow cast by both poles are at the same time of day, we know that angles C and F are equivalent. As a result, angles A and D must also be equivalent.
Since these are similar triangles, we can set up proportions for the corresponding sides.
Now, solve for
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In the above triangle, if 
and 
what is the area of the triangle?
In the above triangle, if
To find the area of a triangle use the formula:
, since the base is 
and the height is 
, plugging in yields:
To find the area of a triangle use the formula:
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The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?
The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?
We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.
We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.
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A right triangle has a total perimeter of 12, and the length of its hypotenuse is 5. What is the area of this triangle?
A right triangle has a total perimeter of 12, and the length of its hypotenuse is 5. What is the area of this triangle?
The area of a triangle is denoted by the equation 1/2 b x h.
b stands for the length of the base, and h stands for the height.
Here we are told that the perimeter (total length of all three sides) is 12, and the hypotenuse (the side that is neither the height nor the base) is 5 units long.
So, 12-5 = 7 for the total perimeter of the base and height.
7 does not divide cleanly by two, but it does break down into 3 and 4,
and 1/2 (3x4) yields 6.
Another way to solve this would be if you recall your rules for right triangles, one of the very basic ones is the 3,4,5 triangle, which is exactly what we have here
The area of a triangle is denoted by the equation 1/2 b x h.
b stands for the length of the base, and h stands for the height.
Here we are told that the perimeter (total length of all three sides) is 12, and the hypotenuse (the side that is neither the height nor the base) is 5 units long.
So, 12-5 = 7 for the total perimeter of the base and height.
7 does not divide cleanly by two, but it does break down into 3 and 4,
and 1/2 (3x4) yields 6.
Another way to solve this would be if you recall your rules for right triangles, one of the very basic ones is the 3,4,5 triangle, which is exactly what we have here
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The length of one leg of an equilateral triangle is 6. What is the area of the triangle?
The length of one leg of an equilateral triangle is 6. What is the area of the triangle?
The base is equal to 6.
The height of an quilateral triangle is equal to 
, where 
is the length of the base.
The base is equal to 6.
The height of an quilateral triangle is equal to
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Find the area of the following right triangle to the nearest integer 
Note: The triangle is not necessarily to scale
Find the area of the following right triangle to the nearest integer
Note: The triangle is not necessarily to scale
The equation used to find the area of a right triangle is: 
where A is the area, b is the base, and h is the height of the triangle. In this question, we are given the height, so we need to figure out the base in order to find the area. Since we know both the height and hypotenuse of the triangle, the quickest way to finding the base is using the pythagorean theorem, 
. a = the height, b = the base, and c = the hypotenuse.
Using the given information, we can write 
. Solving for b, we get 
or 
. Now that we have both the base and height, we can solve the original equation for the area of the triangle. 
The equation used to find the area of a right triangle is:
Using the given information, we can write
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A right triangle has integer sides with a ratio of 
, measured in 
. What is the smallest possible area of this triangle?
A right triangle has integer sides with a ratio of
The easiest way to find the smallest possible integer sides is to simply factor the ratio we are given. In this case, 
is already prime (since 
is a prime number), so the smallest possible sides which hold to this triangle are 
and 
. You may also recognize this number as a common Pythagorean triple.
The area of a triangle is expressed as 
, where 
is the length and 
is the height. Since our triangle is right, we know that two lines intersect at a 
angle and thus serve well as our length and height. We also know that the longest side is always the hypotenuse, so the other two sides must be 
and 
.
Applying our formula, we get:
Thus, the smallest possible area for our triangle is 
.
The easiest way to find the smallest possible integer sides is to simply factor the ratio we are given. In this case,
The area of a triangle is expressed as
Applying our formula, we get:
Thus, the smallest possible area for our triangle is
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Right triangle 
has hypotenuse 
cm and 
. Find the area of the triangle, in cm2, by using 
.
Round angles to four significant figures. Round side lengths to the nearest integer.
Right triangle
Round angles to four significant figures. Round side lengths to the nearest integer.
To find the area of a right triangle, find the lengths of the two perpendicular legs (since this gives us our length and height for the area formula).
In this case, we know that one angle is 
, and SOHCAHTOA tells us that 
, so:
Substitute the angle measure and hypotenuse into the formula.
Isolate the variable.
Solve the left side (rounding to the nearest integer) using our Pythagorean formula:
---> 
Substitute known values.
---> Simplify.
Square root both sides.
So with our two legs solved for, we now only need to apply the area formula for triangles to get our answer:
So, the area of our triangle is 
.
To find the area of a right triangle, find the lengths of the two perpendicular legs (since this gives us our length and height for the area formula).
In this case, we know that one angle is
So with our two legs solved for, we now only need to apply the area formula for triangles to get our answer:
So, the area of our triangle is
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Find the area of a right triangle whose height is 4 and base is 5.
Find the area of a right triangle whose height is 4 and base is 5.
To solve, simply use the formula for the area of a triangle given height h and base B.
Substitute
and 
into the area formula.
Thus,
To solve, simply use the formula for the area of a triangle given height h and base B.
Substitute
Thus,
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