KutaSoftware: Geometry- Similar Triangles Part 3 - Free Printable

Let’s solve this step by step.

We are told that the triangles are similar. That means their sides are proportional — so we can set up a proportion using corresponding sides.

Looking at the diagram:

- The big triangle has height 72 and base 64.
- Inside it, there’s a smaller triangle (or part of it) with height 36 and base labeled as “–4 + 4x”.
- Also, another side is given as 96 — which seems to be the hypotenuse or slanted side of the larger shape? Wait — actually, looking again, the 96 is likely the full slanted side of the large triangle, and 36 is the slanted side of the small triangle inside.

But in the handwritten work, they wrote:

> 96 / 64 = 36 / (–4 + 4x)

That suggests they’re setting up a proportion between two pairs of corresponding sides:

- One pair: 96 (large slanted side) and 64 (large base)
- Other pair: 36 (small slanted side) and (–4 + 4x) (small base)

Wait — that doesn’t make sense for similar triangles. In similar triangles, you match corresponding sides — like base to base, height to height, slant to slant.

Actually, let’s think differently.

The vertical side on the left is 72 total, and from J to T is 27, so from T down to bottom must be 72 – 27 = 45? But that’s not used.

Alternatively, maybe the two triangles sharing the same angle at K? Or perhaps the small triangle on top (with base JU = –4 + 4x and height 27?) and the whole triangle?

Wait — look at the numbers:

They have:

Large triangle: base = 64, slant side = 96

Small triangle: base = –4 + 4x, slant side = 36

And they set up:

96 / 64 = 36 / (–4 + 4x)

This implies they think the ratio of slant to base is the same in both triangles — which would be true if the triangles are similar and those are corresponding sides.

So let’s go with that setup since it’s what’s written and probably intended.

So:

96 / 64 = 36 / (–4 + 4x)

First, simplify 96/64.

Both divisible by 32: 96 ÷ 32 = 3, 64 ÷ 32 = 2 → so 3/2

Or divide numerator and denominator by 16: 96÷16=6, 64÷16=4 → 6/4 = 3/2

So:

3/2 = 36 / (–4 + 4x)

Now cross-multiply:

3 * (–4 + 4x) = 2 * 36

Calculate right side: 2 * 36 = 72

Left side: 3*(–4) + 3*(4x) = –12 + 12x

So:

–12 + 12x = 72

Add 12 to both sides:

12x = 84

Divide by 12:

x = 7

Wait — but let’s check if this makes sense.

Plug x = 7 into –4 + 4x:

–4 + 4*7 = –4 + 28 = 24

Then check proportion:

96 / 64 = 1.5

36 / 24 = 1.5 → yes, matches!

So x = 7 is correct.

But wait — in the original problem, is the small triangle really having base –4 + 4x and slant 36? And large triangle base 64, slant 96?

If so, then yes, the proportion holds.

Alternatively, maybe the heights should be used? Let’s see.

Total height is 72, and from J to T is 27 — so if the small triangle is on top, its height might be 27, and large triangle height 72.

Then ratio of heights: 27/72 = 3/8

Then bases should also be in ratio 3/8.

So small base / large base = 3/8

Small base = –4 + 4x

Large base = 64

So:

(–4 + 4x)/64 = 3/8

Multiply both sides by 64:

–4 + 4x = (3/8)*64 = 3*8 = 24

Then:

4x = 24 + 4 = 28

x = 7

Same answer!

Also, slant sides: if small slant is 36, large slant is 96, ratio 36/96 = 3/8 — same as height ratio.

So all consistent.

Therefore, x = 7.

Final Answer:
7