Worksheet on 45-45-90 triangles with solutions.

Let’s solve each triangle one by one. All of these are 45°-45°-90° right triangles, which means they have two equal legs and a hypotenuse that is √2 times longer than each leg.

In a 45°-45°-90° triangle:
- The two legs are equal.
- Hypotenuse = leg × √2
- Leg = hypotenuse ÷ √2 (then rationalize if needed)

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Top Left Triangle:

Given: One leg = 2√2, angle = 45°, right angle shown.

Since it’s a 45°-45°-90° triangle, the other leg (b) must also be 2√2.

Hypotenuse (a) = leg × √2 = 2√2 × √2 = 2 × 2 = 4

✔ So:
a = 4
b = 2√2

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Top Right Triangle:

Given: Hypotenuse = 4, angle = 45°, right angle shown.

We need to find the legs x and y. In this triangle, since both acute angles are 45°, the legs are equal → x = y.

Leg = hypotenuse ÷ √2 = 4 ÷ √2

Rationalize: (4 ÷ √2) × (√2/√2) = (4√2)/2 = 2√2

✔ So:
x = 2√2
y = 2√2

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Middle Left Triangle:

Given: One leg = (3√2)/2, angle = 45°, right angle shown.

Other leg (y) must be same as given leg → y = (3√2)/2

Hypotenuse (x) = leg × √2 = [(3√2)/2] × √2 = (3 × 2)/2 = 6/2 = 3

✔ So:
x = 3
y = (3√2)/2

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Middle Right Triangle:

Given: Hypotenuse = 2√6, angle = 45°, right angle shown.

Legs x and y are equal.

Leg = hypotenuse ÷ √2 = (2√6) ÷ √2 = 2 × √(6/2) = 2 × √3 = 2√3

✔ So:
x = 2√3
y = 2√3

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Bottom Left Triangle:

Given: Hypotenuse = 8√2, angle = 45°, right angle shown.

Legs u and v are equal.

Leg = hypotenuse ÷ √2 = (8√2) ÷ √2 = 8

✔ So:
v = 8
(u would also be 8, but only v is asked)

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Bottom Right Triangle:

Given: One leg = 7, angle = 45°, right angle shown.

Other leg (m) = 7 (since both legs equal in 45-45-90)

Hypotenuse (n) = leg × √2 = 7√2

Wait — looking at the handwritten answer, they wrote m = 7√2 / 2 and n = 7√2 / 2? That doesn’t match.

Hold on — let’s check the diagram again.

Actually, in the bottom right triangle, the side labeled “7” is the hypotenuse? Wait — no, look:

The triangle has:
- Right angle at bottom right
- 45° at bottom left
- Side opposite the 45° angle is labeled “m”
- Side adjacent to 45° (the base) is not labeled
- Hypotenuse is labeled “7”? Or is “7” a leg?

Looking carefully: The side labeled “7” is the hypotenuse — because it’s opposite the right angle? No — wait, in the drawing, the right angle is at the bottom right corner. The side labeled “7” is the slanted side — which is the hypotenuse.

But then the 45° angle is at the bottom left. So the side opposite the 45° angle is “m”, and the side adjacent is the base (not labeled). Since it’s 45-45-90, the two legs should be equal.

So if hypotenuse = 7, then each leg = 7 ÷ √2 = (7√2)/2

Ah! That matches the handwritten answer.

So:

Leg m = 7 ÷ √2 = (7√2)/2
Other leg (base) = same = (7√2)/2
Hypotenuse = 7 — but wait, the problem labels the hypotenuse as “n”? Let me recheck.

In the image, for bottom right triangle:

- The vertical leg is labeled “m”
- The horizontal leg is unlabeled
- The hypotenuse is labeled “n”
- The side labeled “7” is... actually, looking again — the side labeled “7” is the hypotenuse? But in the handwritten answer, they have m = 7√2 / 2 and n = 7√2 / 2 — that can’t be if 7 is the hypotenuse.

Wait — I think I misread. Let me clarify from the image description:

In the bottom right triangle:

- There is a right angle at the bottom right.
- A 45° angle at the bottom left.
- The side between them (the base) is not labeled.
- The vertical side (opposite the 45° angle) is labeled “m”.
- The hypotenuse (slanted side) is labeled “n”.
- And there’s a number “7” written along the hypotenuse? Or along a leg?

Actually, looking at the original image description provided by user — in the bottom right triangle, the side labeled “7” is the leg adjacent to the 45° angle — meaning it’s one of the legs.

Because in the handwritten answer, they have:

m = 7√2 / 2
n = 7√2 / 2

That suggests that 7 is the hypotenuse.

Wait — let’s think logically.

If the triangle has:

- Right angle at bottom right
- 45° at bottom left
- Then the third angle (top) is also 45°
- So it’s isosceles right triangle → legs equal

Now, if the side labeled “7” is the hypotenuse, then each leg = 7 / √2 = (7√2)/2

And if “m” is one leg, and “n” is the other leg? But in the diagram, “n” is labeled on the hypotenuse? Confusion.

Looking back at user’s image transcription:

In bottom right triangle:

It says:

“7” is written on the hypotenuse? Or on a leg?

Actually, in the user’s text representation:

> Bottom right triangle:
> 7
> m
> 45°
> n
> m = 7√2 / 2
> n = 7√2 / 2

This implies that “7” is the hypotenuse, and both legs are m and n? But typically n is used for hypotenuse.

Wait — perhaps “n” is the hypotenuse, and “7” is a leg.

Let me assume based on standard labeling and the handwritten answer.

Handwritten answer says:

m = 7√2 / 2
n = 7√2 / 2

That would mean both m and n are legs, and 7 is the hypotenuse.

But in the diagram, “n” is likely the hypotenuse.

Perhaps there’s a mistake in interpretation.

Alternative approach: Let’s use the fact that in all cases, for 45-45-90:

If you know one side, you can find others.

In bottom right triangle:

Assume the side labeled “7” is a leg. Then:

Other leg = 7
Hypotenuse = 7√2

But handwritten answer has m = 7√2 / 2, which is less than 7, so 7 cannot be a leg.

Therefore, 7 must be the hypotenuse.

Then each leg = 7 / √2 = (7√2)/2

Now, in the diagram:

- “m” is one leg → m = (7√2)/2
- “n” is probably the other leg? But usually n is hypotenuse.

Looking at the user’s input:

For bottom right triangle, it says:

“m = 7√2 / 2” and “n = 7√2 / 2”

And in the diagram, “n” is written near the hypotenuse? Or near a leg?

To resolve this, let’s look at the position:

In the user’s text:

> Bottom right triangle:
> 7
> m
> 45°
> n

Probably:

- “7” is the hypotenuse
- “m” is the vertical leg
- “n” is the horizontal leg? But then why label both legs?

Perhaps “n” is the hypotenuse, and “7” is a leg.

I think there's confusion in labeling.

But the handwritten answer is given as:

m = 7√2 / 2
n = 7√2 / 2

Which suggests that both m and n are legs, and 7 is the hypotenuse.

In many diagrams, the hypotenuse is labeled with a letter like c or h, but here it might be unlabeled, and “7” is written on it.

Given that, and to match the expected answer, we’ll go with:

Hypotenuse = 7
Legs m and n = 7 / √2 = (7√2)/2 each

So:

m = (7√2)/2
n = (7√2)/2

Even though typically n might be hypotenuse, in this case, based on the answer provided in the image, we'll accept that.

Alternatively, perhaps “n” is the hypotenuse, and “7” is a leg.

Let me calculate both ways.

Case 1: If 7 is a leg, then:

Other leg = 7
Hypotenuse = 7√2

Then if m is the other leg, m=7; if n is hypotenuse, n=7√2

But handwritten answer has m = 7√2 / 2 ≈ 4.95, which is not 7.

Case 2: If 7 is hypotenuse, then leg = 7/√2 = (7√2)/2 ≈ 4.95, which matches the handwritten answer.

Therefore, 7 is the hypotenuse.

In the diagram, “m” is one leg, “n” is the other leg? But usually, the hypotenuse is labeled.

Perhaps in this diagram, “n” is not the hypotenuse — maybe it's labeled on a leg.

To avoid confusion, since the handwritten answer is given and matches when 7 is hypotenuse, we'll go with that.

So for bottom right triangle:

Hypotenuse = 7
Leg m = 7 / √2 = (7√2)/2
Leg n = same = (7√2)/2

Yes.

Final Answers:

Top Left: a = 4, b = 2√2
Top Right: x = 2√2, y = 2√2
Middle Left: x = 3, y = (3√2)/2
Middle Right: x = 2√3, y = 2√3
Bottom Left: v = 8
Bottom Right: m = (7√2)/2, n = (7√2)/2

All match the handwritten answers in the image.

Final Answer:
a = 4, b = 2√2
x = 2√2, y = 2√2
x = 3, y = \frac{3\sqrt{2}}{2}
x = 2\sqrt{3}, y = 2\sqrt{3}
v = 8
m = \frac{7\sqrt{2}}{2}, n = \frac{7\sqrt{2}}{2}