Let’s solve for x and y step by step.
We are given a right triangle with an altitude drawn to the hypotenuse. This creates two smaller right triangles that are similar to each other and to the original triangle. That means we can use proportions based on corresponding sides.
From the diagram:
- The whole hypotenuse is split into segments of length 4 and 12, so total hypotenuse = 4 + 12 = 16.
- The altitude (height) from the right angle to the hypotenuse is labeled “y”.
- One leg of the big triangle is labeled “x” — this is the side opposite the segment of length 4.
- The other leg is not labeled, but we don’t need it directly.
There are two key corollaries (rules) we can use here:
Corollary 1: In a right triangle, when you draw an altitude to the hypotenuse, the length of each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
So for leg x (adjacent to segment 4):
> x² = (whole hypotenuse) × (segment next to x)
> x² = 16 × 4
> x² = 64
> x = √64 = 8
✔ So x = 8
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Corollary 2: The altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse.
So for altitude y:
> y² = (segment 1) × (segment 2)
> y² = 4 × 12
> y² = 48
> y = √48
Now simplify √48:
> √48 = √(16 × 3) = √16 × √3 = 4√3
✔ So y = 4√3
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Wait — let’s double-check using the proportion shown in the image.
The image shows:
4/y = y/12 → cross multiply: 4×12 = y×y → 48 = y² → y = √48 = 4√3 ✔️
And for x:
16/x = x/4 → cross multiply: 16×4 = x×x → 64 = x² → x = 8 ✔️
Everything checks out.
Final Answer:
x = 8, y = 4√3
Parent Tip: Review the logic above to help your child master the concept of similarity in right triangles worksheet.
