The ratio of the legs of a right-angled triangle is three-sevenths, and the length of the height of the line drawn

| April 11, 2021 | education

The ratio of the legs of a right-angled triangle is three-sevenths, and the length of the height of the line drawn from the top of the right angle is 42 cm. Find the length of the segments into which this height divides the hypotenuse.

1. Vertices of the triangle A, B, C. ∠C = 90 °. Height CE = 42 centimeters. BC: AC = 3: 7.

2. AC = 7BC / 3 centimeters.

3. We take the length of the BC leg as x (centimeters). AC = 7x / 3 centimeters.

4. AB² = BC² + AC² (by the Pythagorean theorem).

AB = √BC² + AC² = √x² (7x / 3) ² = √x² + (7x / 3) ² = x√58 / 3 centimeters.

4. According to the properties of a right-angled triangle, the CE height is calculated by the formula:

CE = BC x AC / AB = (7x² / 3) (3 / (x√58) = 7x / √58 = 42.

x = 6√58 centimeters. BC = 6√58 centimeters. AC = 6√58 x 7/3 = 14√58 centimeters.

5. BE = √BC² – CE² = √ (6√58) ² – 42² = √36 x 58 – 1764 = √324 = 18 centimeters.

6. AE = √AC² – CE² = √ (14√58) ² – 42² = √11368 – 1764 = √324 = 98 centimeters.

Answer: AE = 98 centimeters, BE = 18 centimeters.



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