Question

PLEASE HELP!!!
In a right triangle ABC, CD is an altitude, such that AD=BC. Find AC, if AB=3 cm, and CD=2 cm.

Answer 1

We are going to use the concept that area of any given triangle can be written as 1/2 * Base * Height .

Area of shown Triangle can be written as = $\frac{1}{2} * AC * BC =\frac{1}{2} * AB * CD$

We know  AB = 3 and CD =2

Lets say  BC = X  so BD =3 - X

In right triangle  BCD,

Using Pythagorean : $(BD)^{2} +(CD)^{2} =(BC)^{2}$

                                 $(3-X)^{2} +(2)^{2} =(X)^{2}$  

                                  $9 -6X + X^{2}+ 4=X^{2}$

                                  X= $\frac{13}{6}$

                                 BC= $\frac{13}{6}$

Now use the Area equation :

$\frac{1}{2} * AC * BC =\frac{1}{2} * AB * CD$

                          AC  * BC   = AB * CD

                         AC *  $\frac{13}{6}$  =  3 * 2

                               AC  = $\frac{36}{13}$ cm    

Answer 2

The length of AC =

$AC=\frac{1}{6} \sqrt{313}$

Further explanation

Triangles are flat fields bounded by 3 intersecting sides and 3 angles

This side can be the same length or different.

If at an angle there is an angle of 90 degrees, this triangle is said to be a right angle

There is a hypotenuse in this triangle which is the sum of the two sides

$\large{\boxed{\bold{c^2=a^2+b^2}}}$

where c = hypotenuse

This formula is known as the Pythagorean theorem which states that:

" the hypotenuse or the longest side in a right triangle equal to the sum of the squares of the other sides".

A right triangle ABC, CD is an altitude, such that AD = BC. AB = 3 cm, and CD = 2 cm.

The CD line divides the ABC triangle into two ADC and BDC triangles

• Let see the BDC triangle

BD = 3-x

CB = x

CD = 2

We use the Pythagorean theorem

CB² = BD² + CD²

x² = (3-x)² + 2²

x² = 9-6x + x² + 4

6x = 13

x = 13/6

So length CB = 13/6

• Let see the ADC triangle

because AD = CB, the length AD = 13/6

CD = 2

We use the Pythagorean theorem

CD² + AD² = AC²

2² + (13/6)² = AC²

4 + 169/36 = AC²

313/36 = AC²

$AC=\frac{1}{6} \sqrt{313}$

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Keywords: Pythagoras' theorem, hypotenuse,altitude, right triangle