How do you find the area of a isosceles triangle with base 10 and perimeter 36?

Question

Noah Atkinson · Accepted Answer

The area is #60# square units

Giving us the perimeter of an isosceles triangle means we are indirectly given the lengths of the three sides. If the base is #10#, the the other two equal sides add to #26#: #36-10 =26# The equal sides are therefore #26 div 2 =13# To find the area of the triangle we need its height. the height of an isosceles triangle can be found from its line of symmetry. If you cut the triangle into two identical triangles, for each we will have: A #90°# angle A base of #5" "rarr (10 div 2)# The hypotenuse is #13# You can therefore use Pythagoras' Theorem to find the third side which will be the height of the triangle. You might recognise that these are two values of the Pythagorean Triple, #5,12,13" "larr# the height is #12#. If not you can work it out. #x^2 = 13^2-5^2# #x^2 = 144# #x = sqrt144# #x=12# Now we know that the height is #12# so we can find the area of the original triangle. #A = (bh)/2# #A = (10 xx12)/2# #A = 60# square units

Caroline Aucoin · Answer

#a = 60#

Let;

#a -> "area"#

#b -> "base"#

#p -> "perimeter"#

#s -> "slant height"#

#h -> "height"#

#p = 36#

#b = 10#

Recall;

#a = 1/2 b xx h#

#p = b + 2s#

Firstly;

#p = b + 2s#

Substituting the values..

#36 = 10 + 2s#

#36 - 10 = 2s#

#26 = 2s

#26/2 = s#

#13 = s#

Therefore, slant height is #13#

Also recall;

Using Pythagoras theorem;

Since we are using pythagoras theorem, the base of the triangle would be halved..

Therefore, #b = 10/2 = 5#

#s^2 = b^2 + h^2#

Substituting the values..

#13^2 = 5^2 + h^2#

#169 = 25 + h^2#

#169 - 25 = h^2#

#144 = h^2#

#sqrt144 = h#

#12 = h#

Hence;

#a = 1/2 b xx h#

Substituting the values..

#a = 1/2 xx 10 xx 12#

#a = 5 xx 12#

#a = 60#

James Atkins · Answer

undefined To find the area of an isosceles triangle with a base of 10 and a perimeter of 36, you first need to determine the lengths of the two equal sides (since it's an isosceles triangle).

Perimeter = Sum of all sides of the triangle
36 = 10 + 2x (where x is the length of each equal side)

From this, you can solve for x:

36 = 10 + 2x
2x = 36 - 10
2x = 26
x = 13

Now that you know the length of the equal sides (which is 13), you can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, since it's an isosceles triangle, the height will be perpendicular to the base and bisect it, forming two right-angled triangles. You can use the Pythagorean theorem to find the height:

Height^2 = x^2 - (1/2 * base)^2
Height^2 = 13^2 - (1/2 * 10)^2
Height^2 = 169 - 25
Height^2 = 144
Height = 12

Now, plug the values into the area formula:

Area = (1/2) * 10 * 12
Area = 60 square units

So, the area of the isosceles triangle is 60 square units.

Sophia Alfred · Answer

undefined To find the area of an isosceles triangle with a base of 10 units and a perimeter of 36 units:

1. Since the triangle is isosceles, it has two equal sides. Let's denote the length of each equal side as $s$ units.

2. We know that the perimeter of a triangle is the sum of the lengths of its three sides. In this case, the perimeter is given as 36 units, so we have:
$$10 + s + s = 36$$

3. Simplifying the equation, we get:
$$10 + 2s = 36$$
$$2s = 36 - 10$$
$$2s = 26$$
$$s = \frac{26}{2}$$
$$s = 13$$

4. Now, we know the length of each equal side ($s$) is 13 units.

5. To find the height of the triangle (the perpendicular distance from the base to the apex), we can use the Pythagorean theorem. Since the triangle is isosceles, the height will split the base into two equal segments.

6. Let $h$ be the height of the triangle. Using the Pythagorean theorem, we have:
$$h^2 + (5)^2 = (13)^2$$
$$h^2 + 25 = 169$$
$$h^2 = 169 - 25$$
$$h^2 = 144$$
$$h = \sqrt{144}$$
$$h = 12$$

7. Now that we have the base and height of the triangle, we can find the area using the formula for the area of a triangle:
$$Area = \frac{1}{2} \times \text{base} \times \text{height}$$
$$Area = \frac{1}{2} \times 10 \times 12$$
$$Area = 60 \text{ square units}$$

Therefore, the area of the isosceles triangle with a base of 10 units and a perimeter of 36 units is 60 square units.