Construct a right triangle whose base is 12cm and sum of its hypotenuse and other side is 18cm

other side = x

hypotenuse = √(x^2 + 12^2)

x + √(x^2 + 144) = 18
√(x^2 + 144) = 18-x
square both sides:
x^2 + 144 = 324 - 36x + x^2
36x = 324-144
x = 5

your triangle is 5 by 12 with a hypotenuse of 13

check:
is 5+13 = 18? YUP
is 5^2 + 12^2 = 13^2 ? , YUP

you should recognize triangles such as 3,4,5 and 5,12,13 as right-angled.
They are the simplest of such triangles with whole numbers.
(note that 6,8,10 is also such a triangle but it is similar to 3,4,5)

To construct a right triangle, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's call the hypotenuse "c" and the other side "a". We are given that the base (which is one of the sides) is 12cm and the sum of the hypotenuse and the other side is 18cm. So, we can translate this information into an equation:

c + a = 18

Since the hypotenuse is always longer than the other two sides, we can solve for the hypotenuse in terms of the other side:

c = 18 - a

Now, we can substitute this into the Pythagorean theorem:

(18 - a)^2 = a^2 + 12^2

Expanding and simplifying:

324 - 36a + a^2 = a^2 + 144

Moving all terms to one side:

0 = 36a - 180

Simplifying further:

36a = 180

Dividing by 36:

a = 5

Now that we know the length of one side, we can substitute it back into our equation to find the length of the hypotenuse:

c = 18 - a
c = 18 - 5
c = 13

So, the other side has a length of 5cm and the hypotenuse has a length of 13cm.

Using this information, we can construct a right triangle by drawing a line segment of 12cm as the base and then drawing two perpendicular lines from the endpoints of the base. The length of the perpendicular lines will be 5cm for one and 13cm for the other, forming a right angle where they meet.