Question

In ΔABC, AC = BC, CD ⊥ AB with D ∈ AB , AB = 4 in, and CD = sqrt3 in. Find AC.

Answer 1

check the picture below.

segments AC and BC are equal, that means that ABC is an isosceles triangle, with twin sides.  Now the twin sides make twin angles on the opposite side, namely the angles at A and B are twins.

if the angles at A and B are twins, and CD is ⟂ to AB, there's only one possibility that can happen, and is if CD is an angle bisector at C.

an angle bisector like CD with twin angles on each side, will cut AB in two equal halves, therefore, if AB = 4, then AD = 2 and DB = 2.

Answer 2

The length of AC is $\sqrt{7}$ inches.

Given in ΔABC, AC = BC it shows that ΔABC is an isosceles triangle.

The length of AB is 4 inches and the length of CD is $\sqrt{3}$ inches.

Since CD is perpendicular to AB, so the ΔABC is divided in two right angled triangle namely ΔADC and ΔBDC.

We have to find the length of AC, so we consider right triangle ADC here.

We know that in right angled triangle,The Pythagoras Theorem states that the sum of the square of the base and square of the perpendicular is equal to the square of the hypotenuse.

So $AC^{2}=AD^{2}+CD^{2}$......equation 1.

In an isosceles triangle the perpendicular drawn from the vertex always bisect the base since here CD ⊥ AB so AD = BD = half of AB = 2.

Now putting the values in equation 1 we get,

$AC^{2}=2^{2} +\sqrt{3} ^{2}$

$AC^{2} =4+3\\AC^{2} =7\\AC=\sqrt{7}$

Hence the length of AC is $\sqrt{7}$ inches.

For more details on Pythagoras theorem follow the link below:

https://brainly.com/question/343682