In the right-angled triangle ABC (angle C = 90, AB = 13, AC = CB-7)

| September 25, 2021 | education

In the right-angled triangle ABC (angle C = 90, AB = 13, AC = CB-7), the bisector CK is drawn. Find the legs of the triangle ABC and the radius of the circle circumscribed about the triangle CKB?

∠C = 90 °, AB = 13, AC = BC – 7.

1. By the Pythagorean theorem:

AC² + BC² = AB²;

(BC – 7) ² + BC² = 13²;

BC² – 14 * BC + 49 + BC² = 169;

2 * BC² – 14 * BC + 49 – 169 = 0;

2 * BC² – 14 * BC – 120 = 0;

BC² – 7 * BC – 60 = 0.

D = (- 7) ² – 4 * 1 * (- 60) = 49 + 240 = 289.

BC₁ = (- (- 7) + √289) / 2 = (7 + 17) / 2 = 24/2 = 12.

BC ₂ = (- (- 7) – √289) / 2 = (7 – 17) / 2 = – 10/2 = – 5 – does not make sense.

Thus:

AC = 12 – 7 = 5.

1. The length of the bisector drawn to side a is found by the formula:

l = 1 / (b + c) * √ (b * c * (a + b + c) * (b + c – a)).

Find the length of CK:

CK = 1 / (AC + BC) * √ (AC * BC * (AB + AC + BC) * (AC + BC – AB)) = (√ (5 * 12 * (13 + 5 + 12) * (5 + 12 – 13))) / (5 + 12) = (√ (60 * 30 * 4)) / 17 = 60√2 / 17.

1. The bisector divides the side to which it is drawn into segments proportional to the adjacent sides:

AK / BK = AC / BC ⇒ AK / BK = 5/12.

AK + BK = 13;

AK = 13 – BK.

Substitute the expression into the relation:

(13 – BK) / BK = 5/12;

5 * BK = 156 – 12 * BK;

17 * BK = 156;

BK = 156/17.

1. Find the area △ CKB by Heron’s formula.

p = (CK + BK + BC) / 2 = (60√2 / 17 + 156/17 + 12) / 2 = (60√2 + 156 + 204) / 17: 2 = (60√2 + 360) / 34 = (30√2 + 180) / 17.

S = √ ((30√2 + 180) / 17 * ((30√2 + 180) / 17 – 60√2 / 17) * ((30√2 + 180) / 17 – 156/17) * (( 30√2 + 180) / 17 – 12)) = √ ((30√2 + 180) / 17 * (- 30√2 + 180) / 17 * (30√2 + 24) / 17 * (30√2 – 24) / 17) = √ ((32400 – 1800) / 289 * (1800 – 576) / 289) = (√ (30600 * 1224)) / 289 = 6120/289 = 360/17.

1. The radius of a circle circumscribed about a triangle is:

R = (a * b * c) / (4 * S).

The radius of a circle circumscribed about △ CKB is:

R = (CK * BK * BC) / (4 * S) = (60√2 / 17 * 156/17 * 12) / (4 * 360/17) = 112320√2 / 289: 1440/17 = 112320√ 2/289 * 17/1440 = 78√2 / 17.

Answer: R = 78√2 / 17.



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