How to find the area of a right triangle - Basic Geometry

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Question

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?

Answer

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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Question

A right triangle has a total perimeter of 12, and the length of its hypotenuse is 5. What is the area of this triangle?

Answer

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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Question

The length of one leg of an equilateral triangle is 6. What is the area of the triangle?

Answer

$A=\frac{1}{2}(B)(H)$

The base is equal to 6.

The height of an quilateral triangle is equal to $\frac{B\sqrt{3}}{2}$ , where is the length of the base.

$A=\frac{1}{2}(6)(\frac{6\sqrt{3}}{2})=\frac{1}{2}(6)(3\sqrt{3})=9\sqrt{3}$

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Question

Given:

A = 3 cm

B = 4 cm

What is the area of the right triangle ABC?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)34 = 6 cm^{2}$

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Question

Given:

A = 4 cm

B = 6 cm

What is the area of the right triangle ABC?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)46 = 12 cm^{2}$

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Question

Given:

A = 3 cm

B = 7 cm

What is the area of the triangle?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)37 = 10.5 cm^{2}$

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Question

Given that:

A = 6 cm

B = 10 cm

What is the area of the right trianlge ABC?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)106 = 12 cm^{2}$

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Question

Given that:

A = 3 cm

B = 4 cm

C = 5 cm

What is the area of the right triangle ABC?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)43 = 6 cm^{2}$

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Question

Given that:

A = 10 cm

B = 20 cm

What is the area of the right triangle ABC?

Answer

The area of a triangle is given by the equation:

$A_{\Delta } = (1/2)(base)(height)$

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

$A_{\Delta }=(1/2)2010 = 100 cm^{2}$

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Question

An equilateral triangle has a side of .

What is the area of the triangle?

Answer

An equilateral triangle has three congruent sides. The area of a triangle is given by $A = \frac{1}{2}bh$ where is the base and is the height.

The equilateral triangle can be broken into two right triangles, where the legs are and and the hypotenuses is .

Using the Pythagorean Theorem we get $4^{2}=2^{2} + x^{2}$ or $x=2\sqrt{3}$ and the area is $A = \frac{1}{2}bh= \frac{1}{2}(4)(2\sqrt{3}) = 4\sqrt{3}$

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Question

The length of the legs of the triangle below (not to scale) are as follows:

cm

cm

What is the area of the triangle?

Answer

The formula for the area of a triangle is

$\frac{1}{2}bh$

where is the base of the triangle and is the height.

For the triangle shown, side is the base and side is the height.

Therefore, the area is equal to

$\frac{1}{2}(7)(12)=42$

or, based on the units given, 42 square centimeters

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Question

The hypotenuse of a $30^{\circ }-60^{\circ }-90^{\circ }$ triangle measures eight inches. What is the area of this triangle (radical form, if applicable)?

Answer

In a $30^{\circ }-60^{\circ }-90^{\circ }$ , the shorter leg is half as long as the hypotenuse, and the longer leg is $\sqrt{3}$ times the length of the shorter. Since the hypotenuse is 8, the shorter leg is 4, and the longer leg is $4\sqrt{3}$ , making the area:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 4 \cdot 4\sqrt{3} = 8\sqrt{3}$

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Question

$What;is;the;area;of;\Delta GHJ?$

Answer

$Area ;\Delta= \frac{1}{2}(base)(height)$

$\Delta GHF=\frac{1}{2}(5)(12)=\frac{1}{2}(60)=30$

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Question

$What;is;the;area;of;\Delta KLM?$

Answer

$Area ;\Delta=\frac{1}{2}(base)(height)$

$Area;\Delta KLM=\frac{1}{2}(3)(4)=\frac{1}{2}(12)=6$

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Question

Find the area of ONE of the triangles formed by the diagonal.

Answer

To find the area of a right triangle, use the formula $A = \frac{1}{2}bh$ , where is the area of the triangle, is the length of the triangle's base, and is the triangle's height.

$A = \frac{1}{2}(8)(12)$

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Question

What is the area of a right triangle with a height of 5 inches and a base of 3 inches?

Answer

To find the area of a triangle, multiply the base by the height, then divide by 2.

$3\cdot 5=15$

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Question

Find the area of the following right triangle to the nearest integer

Note: The triangle is not necessarily to scale

Answer

The equation used to find the area of a right triangle is: $A = \frac{1}{2}b\cdot h$ 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, $a^{2}+b^{2}=c^{2}$ . a = the height, b = the base, and c = the hypotenuse.

Using the given information, we can write $8^{2} + b^{2} = 13^{2}$ . Solving for b, we get $b^{2} = 105$ or $b \approx 10.25$ . Now that we have both the base and height, we can solve the original equation for the area of the triangle. $A = \frac{1}{2}\cdot10.25\cdot8 = 40.99\approx41$

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Question

Find the area of this right triangle:

Answer

The area of a triangle is $\small \frac{1}{2}(base)(height)$ . For a right triangle, we can use the two legs as the base and the height regardless of orientation. Right now we only know one of the legs, and we can solve for the other using Pythagorean Theorem:

$\small \small 4^2 + x^2 = 7^2$

$\small 16 + x^2 = 49$ subtract 16 from both sides

$\small x^2 = 33$

$\small x = \sqrt{33}$ we can see that all of the answers are in radical form, so we do not have to simplify this to a crazy decimal.

Now we have our "base" and "height" and we can plug them into the formula to find the area:

$\small \frac{1}{2}4\sqrt{33}$ half of 4 is 2, so our answer is $\small 2\sqrt{33}$ , which we can't simplify any more.

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Question

Find the area of this right triangle:

Answer

The area of a triangle is found using the formula $\small \frac{1}{2}(base)(height)$ . For a right triangle, we can use the two legs as the base and height regardless of the orientation of the triangle.

In this case, we only know one leg, but we can figure out the other by using the Pythagorean Theorem:

$\small 6^2 + x^2 = 10^2$

$\small 36 + x^2 = 100$ subtract 36 from both sides

$\small x^2 = 64$ take the square root

$\small x = \sqrt{64 } = 8$

Now that we have our "base" and "height," we can substitute them into the formula and find the area:

$\small \frac{1}{2}86 = 4*6 = 24$

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Question

Find the area of a right triangle whose perpendicular sides are of lengths and .

Answer

To find area of this triangle, you must use the formula indicated below.

$A=\frac{1}{2}bh$

where is base and is height. Thus:

$A=\frac{1}{2}(5)(4)=10$

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