What is the height of a triangle if the base is 10 cm and the sides are 13 cm

If the sides (plural, meaning more than 1) and the base is 10

that means
2 sides=13 cm
base=10
since 2 sides equal legnths, isocolees
draw a line from the top angle, opposite the base side, perpendicular to the base
you will form 2 right triangles each with a base of 10/2=5 and a hypotonuse of 13 solve for other leg

a^2+b^2=c^2
c=hypotonuse=13
a=10=1 leg
b=mystery leg (height)

10^2+x^2=13^2
100+x^2=169
subtract 100 from both sides
x^2=69
take square root of both sides
x=√69

it would be safe to put √69 as the answe since it cannot be simplified into a whole number (about 8.3066238629181)

Because the sides are the same it is an isosceles triangle, therefore the altitude forms two right triangles and also bisects the base

Apply the Pythagorean Theorem
5^2 + h^2= 13^2
 
25+h^2= 169
     h^2= 144
       h=12

FYI  there are familiar combinations for right triangles, might be worthwhile to check them out, it will save a lot of time.

Some examples are 3,4,5  /  5,12. 13  / 9,40,41 etc

Hope this helps