Question

In right △ABC altitude overlineBD is drawn to hypotenuse overline{AC} If AD = 5 and BC = 2√ 21, find DC, BD, and ABAB.

Answer 1

`\overline {DC}` = 7

`\overline {BD}` = √35

`\overline {AB}` = 2·√15

Explanation:

Please find attached the drawing of right triangle ABC created with MS Visio

Let 'x' represent
`\overline {BD}` let 'y' represent
`\overline {DC}`, and let 'z' represent
`\overline {AB}`, we get;

5² + x² = z²...(1)

x² + y² = (2·√21)² = 84

x² + y² = 84...(2)

84 + z² = (5 + y)²...(3)

From equation (1), we have;

x² = z² - 5²...(4)

From equation (2), and equation (4), we have;

x² + y² = 84 = (z² - 5²) + y² = 84

∴ z² = 84 - y² + 5² = 109 - y²

z² = 109 - y²...(5)

From equation (3), and equation (5), we have;

84 + z² = (5 + y)² = 84 + (109 - y²) = (5 + y)²

84 + (109 - y²) = (5 + y)²

(5 + y)² - (84 + (109 - y²)) = 0

2·y² + 10·y - 168 = 0

y² + 5·y - 84 = 0

y = (-5 ± √(5² - 4 × 1 × (-84)))/(2 × 1)

y = 7, or y = -12

We note that 'y' is a natural number, therefore, the correct option of the two options is y = 7

y =
`\overline {DC}` = 7

From equation (2), we have;

x² + y² = 84

x² = 84 - y² = 84 - 7² = 35

x = √35

x =
`\overline {BD}` = √35

z² = 109 - y²

∴ z² = 109 - 7² = 60

z = √60 = 2·√15

z = 2·√15

z =
`\overline {AB}` = 2·√15.