Practice problems for finding the value of x in 45-45-90 special right triangles.

Let’s solve each triangle one by one. These are all 45-45-90 right triangles, which means:

→ The two legs (the sides that form the right angle) are equal in length.
→ The hypotenuse (the side opposite the right angle) is always leg × √2.

We can use this rule to find x in each case.

---

Problem 1:

Given: Hypotenuse = 4√2, and it’s a 45-45-90 triangle.
Solve for: leg = x

In a 45-45-90 triangle:
Hypotenuse = leg × √2
So:
4√2 = x × √2
Divide both sides by √2:
x = 4

✔ Answer: x = 4

---

Problem 2:

Given: One leg = 6, angle = 45°, so it’s a 45-45-90 triangle.
Solve for: hypotenuse = x

Hypotenuse = leg × √2
x = 6 × √2
x = 6√2

✔ Answer: x = 6√2

---

Problem 3:

Given: Hypotenuse = 2√2
Solve for: leg = x

Again:
Hypotenuse = leg × √2
2√2 = x × √2
Divide both sides by √2:
x = 2

✔ Answer: x = 2

---

Problem 4:

Given: One leg = 8, angle = 45° → 45-45-90 triangle
Solve for: hypotenuse = x

x = 8 × √2
x = 8√2

✔ Answer: x = 8√2

---

Problem 5:

Given: One leg = 10, angle = 45° → 45-45-90 triangle
Solve for: hypotenuse = x

x = 10 × √2
x = 10√2

✔ Answer: x = 10√2

---

Problem 6:

Given: One leg = 3, angle = 45° → 45-45-90 triangle
But note: the right angle is at the top, and the 45° angle is at the bottom right — so the side labeled “3” is adjacent to the 45° angle, meaning it’s a leg. The side labeled “x” is the other leg (since it’s also next to the right angle). In a 45-45-90 triangle, both legs are equal!

Wait — let’s double-check the diagram description.

Actually, looking again:
The triangle has:
- Right angle at top right corner.
- 45° angle at bottom right.
- So the side labeled “3” is the vertical leg (from bottom right up to top right).
- The side labeled “x” is the horizontal leg (from top left to top right).

Since it’s a 45-45-90 triangle, the two legs must be equal. So if one leg is 3, the other leg (x) must also be 3.

BUT — wait! Is “x” really a leg? Let me re-read.

Actually, in problem 6, the side labeled “x” is the top side — which is between the right angle and the 45° angle? No — let’s think carefully.

If the right angle is at the top right, and the 45° angle is at the bottom right, then:

- The side from bottom right to top right is vertical → length 3 → this is one leg.
- The side from top right to top left is horizontal → labeled x → this is the other leg.
- The hypotenuse is from top left to bottom right.

Since it’s a 45-45-90 triangle, the two legs are equal. So x = 3.

✔ Answer: x = 3

Wait — but what if “x” was meant to be the hypotenuse? Let’s check the labeling.

Looking back at the original image description (even though we’re not describing it), in problem 6, the side labeled “x” is the top side, which is adjacent to the right angle and not opposite the 45° angle — so yes, it’s a leg. And since the other leg is 3, and angles are 45-45-90, legs are equal → x = 3.

Alternatively, if someone misreads and thinks x is the hypotenuse, they’d get 3√2 — but based on standard labeling and position, x is the other leg.

To confirm: in a 45-45-90 triangle, if you know one leg, the other leg is the same, and hypotenuse is leg×√2.

Here, given leg = 3, and x is the other leg → x = 3.

✔ Final confirmation: x = 3

---

Now, compiling all answers:

Final Answer:
1. x = 4
2. x = 6√2
3. x = 2
4. x = 8√2
5. x = 10√2
6. x = 3