How to find the perimeter of a right triangle - SAT Math

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Question

Based on the information given above, what is the perimeter of triangle ABC?

Answer

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

$17.5+2(17.5)+17.5\sqrt{3}=52.2+17.5\sqrt{3}$

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Question

Three points in the xy-coordinate system form a triangle.

The points are .

What is the perimeter of the triangle?

Answer

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is $\sqrt{41}$ .

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Question

What is the perimeter of the triangle above?

Answer

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg $\overline{BC}$ is equal to that of hypotenuse $\overline{AC}$ , the length of which is 12, divided by $\sqrt{2}$ . Therefore,

$BC = \frac{AC}{\sqrt{2}} = \frac{12}{\sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$BC = \frac{12 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2}$

By the same reasoning, $AB= 6 \sqrt{2}$ .

The perimeter of the triangle is

$P= 6 \sqrt{2} + 6 \sqrt{2} + 12 = 12+ 12 \sqrt{2}$

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Question

Give the perimeter of the provided triangle.

Answer

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle.

By the 45-45-90 Triangle Theorem, legs $\overline{AB}$ and $\overline{BC}$ are of the same length, so

.

Also by the 45-45-90 Triangle Theorem, the length of hypotenuse $\overline{AC}$ is equal to that of leg $\overline{BC}$ multiplied by $\sqrt{2}$ . Therefore,

$AC = AB \cdot \sqrt{2} = 12 \sqrt{2}$ .

The perimeter of the triangle is

$P = AB + B C + AC = 12+ 12+ 12 \sqrt{2} = 24 + 12 \sqrt{2}$

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