Question

the triangles are similar
what is the value of x?
enter your answer in the box.
x =
image of two right triangles, larger one with legs 72 and x, hypotenuse 75; smaller one with legs 24 and 7, hypotenuse 25

Explanation:

Step1: Recall similar triangles property

For similar triangles, corresponding sides are proportional. Let's identify the corresponding sides. The hypotenuses are 75 and 25, the vertical sides are 72 and 24, and the horizontal sides are \( x \) and 7. First, find the scale factor. The scale factor from the smaller triangle to the larger triangle is \( \frac{75}{25}=3 \) (or using vertical sides \( \frac{72}{24}=3 \)).

Step2: Apply scale factor to find \( x \)

Since the horizontal side of the smaller triangle is 7, and the scale factor is 3, we multiply 7 by 3 to get \( x \). So \( x = 7\times3 = 21 \)? Wait, no, wait. Wait, maybe I mixed up. Wait, let's check the sides. Wait, the vertical side of the larger triangle is 72, smaller is 24. So ratio of larger to smaller is \( \frac{72}{24}=3 \). So the horizontal side of larger (x) should be 3 times the horizontal side of smaller (7). Wait, but wait, let's check the hypotenuse: 75 and 25, ratio 3. So yes, scale factor 3. So \( x = 7\times3 = 21 \)? Wait, no, wait, maybe I got the corresponding sides wrong. Wait, the right triangles: the legs are (72, x) and (24,7), hypotenuses 75 and 25. So the ratio of legs: 72/24 = 3, 75/25 = 3, so the other leg should be 7*3=21? Wait, but let's verify with Pythagoras. For the smaller triangle: 24² +7² = 576 +49=625=25², correct. For the larger triangle: 72² +x² should be 75². 75²=5625, 72²=5184. So x²=5625 -5184=441, so x=21. Yes, that's correct. So step by step:

First, confirm similarity: corresponding angles equal (right angles and same shape), so sides proportional.

Let’s set up the proportion: \( \frac{72}{24}=\frac{x}{7} \) (since vertical sides are corresponding, horizontal sides are corresponding).

So \( \frac{72}{24}=\frac{x}{7} \)

Simplify \( \frac{72}{24}=3 \), so \( 3=\frac{x}{7} \)

Multiply both sides by 7: \( x = 3\times7 = 21 \)

Answer:

21