Let , and denote the length of the sides of a right triangle, with being the length of the hypotenuse. Find the missing sidelength in each of the following cases: (i) (ii) (iii) (iv) (v)

We will use the Pythagorean theorem to find the missing side length. This theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If and are the lengths of the two shorter sides, and is the length of the hypotenuse, then . Step 1 — Finding when We are given the lengths of the two shorter sides. Let and . We need to find the length of the hypotenuse, . Let us apply the Pythagorean theorem. > Step 2 — Finding when Here, we have the shorter sides and . We need to find the hypotenuse, . Let us use the Pythagorean theorem. We can simplify the square root of 208. Let us find the prime factors of 208. . > Step 3 — Finding when In this case, we know one shorter side . We also know the hypotenuse . We need to find the other shorter side, . From the Pythagorean theorem, . We can rearrange this to find . > Step 4 — Finding when We are given the shorter sides and . We need to find the hypotenuse, . Let us use the Pythagorean theorem. The number 193 is a prime number. So, its square root cannot be simplified further. > Step 5 — Finding when Here, we have decimal side lengths. Let and . We need to find the hypotenuse, . Let us apply the Pythagorean theorem. > Answer > (i) > (ii) > (iii) > (iv) > (v)

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Question 7

Let $a$ , $b$ and $c$ denote the length of the sides of a right triangle, with $c$ being the length of the hypotenuse. Find the missing sidelength in each of the following cases:

(i) $a = 5, b = 7$

(ii) $a = 8, b = 12$

(iii) $a = 9, c = 15$

(iv) $a = 7, b = 12$

(v) $a = 1.5, b = 3.5$

Solution

We will use the Pythagorean theorem to find the missing side length. This theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If $a$ and $b$ are the lengths of the two shorter sides, and $c$ is the length of the hypotenuse, then $a^2 + b^2 = c^2$ .

**Step 1 — Finding $c$ when $a=5, b=7$ }

We are given the lengths of the two shorter sides. Let $a = 5$ and $b = 7$ . We need to find the length of the hypotenuse, $c$ . Let us apply the Pythagorean theorem.

$c^2 = a^2 + b^2$

$c^2 = 5^2 + 7^2$

$c^2 = 25 + 49$

$c^2 = 74$

$c = \sqrt{74}$

$\boxed{c = \sqrt{74}}$

Diagram 1

**Step 2 — Finding $c$ when $a=8, b=12$ }

Here, we have the shorter sides $a = 8$ and $b = 12$ . We need to find the hypotenuse, $c$ . Let us use the Pythagorean theorem.

$c^2 = a^2 + b^2$

$c^2 = 8^2 + 12^2$

$c^2 = 64 + 144$

$c^2 = 208$

$c = \sqrt{208}$

We can simplify the square root of 208. Let us find the prime factors of 208. $208 = 16 \times 13$ .

$c = \sqrt{16 \times 13}$

$c = \sqrt{16} \times \sqrt{13}$

$c = 4 \times \sqrt{13}$

$\boxed{c = 4\sqrt{13}}$

Diagram 2

**Step 3 — Finding $b$ when $a=9, c=15$ }

In this case, we know one shorter side $a = 9$ . We also know the hypotenuse $c = 15$ . We need to find the other shorter side, $b$ . From the Pythagorean theorem, $a^2 + b^2 = c^2$ . We can rearrange this to find $b^2$ .

$b^2 = c^2 - a^2$

$b^2 = 15^2 - 9^2$

$b^2 = 225 - 81$

$b^2 = 144$

$b = \sqrt{144}$

$b = 12$

$\boxed{b = 12}$

Diagram 3

**Step 4 — Finding $c$ when $a=7, b=12$ }

We are given the shorter sides $a = 7$ and $b = 12$ . We need to find the hypotenuse, $c$ . Let us use the Pythagorean theorem.

$c^2 = a^2 + b^2$

$c^2 = 7^2 + 12^2$

$c^2 = 49 + 144$

$c^2 = 193$

$c = \sqrt{193}$

The number 193 is a prime number. So, its square root cannot be simplified further.

$\boxed{c = \sqrt{193}}$

Diagram 4

**Step 5 — Finding $c$ when $a=1.5, b=3.5$ }

Here, we have decimal side lengths. Let $a = 1.5$ and $b = 3.5$ . We need to find the hypotenuse, $c$ . Let us apply the Pythagorean theorem.

$c^2 = a^2 + b^2$

$c^2 = (1.5)^2 + (3.5)^2$

$c^2 = 2.25 + 12.25$

$c^2 = 14.5$

$c = \sqrt{14.5}$

$\boxed{c = \sqrt{14.5}}$

Diagram 5

Answer

(i) $c = \sqrt{74}$ (ii) $c = 4\sqrt{13}$ (iii) $b = 12$ (iv) $c = \sqrt{193}$ (v) $c = \sqrt{14.5}$

Question 7

Answer

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