Let △𝐴𝐵𝐶
△
A
B
C
be a right triangle with ∠𝐴=90∘
∠
A
=
90
∘
. Suppose ∠𝐵=45∘
∠
B
=
45
∘
and 𝐵𝐶=152‾√
B
C
=
15
2
. Find 𝐴𝐵
A
B
.

We are given a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), \( \angle B = 45^\circ \), and \( BC = 15\sqrt{2} \). We are tasked with finding \( AB \).

Since \( \angle B = 45^\circ \) and the sum of angles in a triangle is \( 180^\circ \), the remaining angle \( \angle C \) must also be \( 45^\circ \). This tells us that \( \triangle ABC \) is an **isosceles right triangle**, where the two legs of the triangle are equal in length.

### Step 1: Properties of Isosceles Right Triangle
In an isosceles right triangle (also known as a 45-45-90 triangle), the legs are equal, and the hypotenuse is \( \sqrt{2} \) times the length of each leg. Therefore, if the hypotenuse is given as \( BC = 15\sqrt{2} \), we can use the following formula for a 45-45-90 triangle:

\[
BC = AB \times \sqrt{2}
\]

### Step 2: Solving for \( AB \)
We can rearrange the equation to solve for \( AB \):

\[
AB = \frac{BC}{\sqrt{2}} = \frac{15\sqrt{2}}{\sqrt{2}} = 15
\]

Thus, the length of \( AB \) is \( 15 \).

### Final Answer:
\[
AB = 15
\]

Would you like more details or further explanation?

Here are 5 related questions for further exploration:
1. How do you derive the properties of a 45-45-90 triangle?
2. What is the length of \( AC \) in this triangle?
3. Can you solve for the area of \( \triangle ABC \)?
4. What is the perimeter of \( \triangle ABC \)?
5. How do the properties of other special triangles (e.g., 30-60-90 triangles) differ from 45-45-90 triangles?

**Tip:** In any right triangle, the sum of the two acute angles is always \( 90^\circ \).

Let △ABC be a right triangle with ∠A = 90°. Suppose ∠B = 45° and BC = 15√2. Find AB.

Learn how to solve a right triangle where ∠A = 90°, ∠B = 45°, and BC = 15√2. This problem explores the properties of 45-45-90 triangles and uses geometric formulas to find the length of AB. Ideal for students in Grades 9-10.

Math Problem Statement

Let △𝐴𝐵𝐶 △ A B C be a right triangle with ∠𝐴=90∘ ∠ A

90 ∘ . Suppose ∠𝐵=45∘ ∠ B

45 ∘ and 𝐵𝐶=152‾√ B C

Solution

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