Question

in right triangle ABC shown in the diagram below, altitude BD is drawn to
hypotenuse AC, CD = 12, and AD
= 3. What is the length of AB?

a. 5 square root 3
b. 6
c. 3 square root 5
d. 9

Answer 1

Answer:

c. This is a geometric mean problem.. so set up the proportion 3/x = x/12. cross multiply to get x^2= 36. take the sq. root of both sides so x =6. x was representing AD. then you can use Pythagorean Theorem to set up 3^2+6^2 = y^2. when you solve for y you get

$\sqrt{45 } = 3 \sqrt{5}$

Answer 2

The length of the side AB in the right triangle ABC is given by: Option C: 3√5 units.

What is Pythagoras Theorem?

If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:

$|AC|^2 = |AB|^2 + |BC|^2$

where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).

For the given case, referring to the image attached below,

Suppose that:

• Length of BD = x units
• Length of AB = y units

Then, for triangle ABC, the length of the hypotenuse is 12 + 3 = 15 units, and perpendicular is |AB| = y units in length, so we get:

$|AC|^2 = |AB|^2 + |BC|^2 \\\\15^2 = y^2 + |BC|^2\\\\|BC| =\sqrt{225 -y^2}$

(positive sq. sides because length cannot be negative).

For triangle ADB, we get:

$|AB|^2= |AD|^2 + |BD|^2\\y^2 = 3^2 + x^2\\y^2 = 9 + x^2$

For triangle CDB, we get:

$|BC|^2= |CD|^2 + |BD|^2\\225-y^2 = 12^2 + x^2\\y^2 = 225 - 144 - x^2 = 81 - x^2$

From both the equations, we get:

$y^2 = 9+x^2 =81- x^2\\9+x^2 = 81- x^2\\2x^2 = 72\\x = \sqrt{36}=6$

(x is length, so cannot be negative 6, thus, only being +6)

Thus, we get:

$y^2 = 81-x^2=81-6^2 = 81-36=45\\y = \sqrt{45} =3\sqrt{5}$

The length of |AB| was denoted by 'y'.

Thus, the length of the side AB in the right triangle ABC is given by: Option C: 3√5 units.

Learn more about Pythagoras theorem here:

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