Respuesta :
we know that
[surface area of a triangular pyramid]=[area of the base]+3*[area of the lateral triangle]
step 1
find the area of the base
is a equilateral triangle
applying the Pythagorean theorem
h²=10²-5²------> h²=100-25----------> h=√75 cm
[area of the base]=10*√75/2---------> 5√75 cm²
step 2
find the area of the lateral triangle
[surface area]=[area of the base]+3*[area of the lateral triangle]
[area of the lateral triangle]={[surface area]-[area of the base]}/3
[area of the lateral triangle]={[214.5]-[5√75]}/3
[area of the lateral triangle]=57.07 cm²
step3
find the slant height
[area of the lateral triangle]=b*[slant height]/2
[slant height]=2*[area of the lateral triangle]/b
[slant height]=2*[57.07]/10-----> [slant height]=11.41 cm
the answer is
the slant height is 11.41 cm
[surface area of a triangular pyramid]=[area of the base]+3*[area of the lateral triangle]
step 1
find the area of the base
is a equilateral triangle
applying the Pythagorean theorem
h²=10²-5²------> h²=100-25----------> h=√75 cm
[area of the base]=10*√75/2---------> 5√75 cm²
step 2
find the area of the lateral triangle
[surface area]=[area of the base]+3*[area of the lateral triangle]
[area of the lateral triangle]={[surface area]-[area of the base]}/3
[area of the lateral triangle]={[214.5]-[5√75]}/3
[area of the lateral triangle]=57.07 cm²
step3
find the slant height
[area of the lateral triangle]=b*[slant height]/2
[slant height]=2*[area of the lateral triangle]/b
[slant height]=2*[57.07]/10-----> [slant height]=11.41 cm
the answer is
the slant height is 11.41 cm
Area of the lateral triangle is half or the product of the slant height and the side length. The slant height of the triangular pyramid is 11.143 centimetres.
Given information-
The side length of the triangular pyramid is 10 centimetres.
The surface area of the triangular pyramid is 214.5 square centimetres.
Slant height-
The slant height is the distance measured along a lateral face from the base to the top along the center of the pyramid.
Surface area of triangular pyramid
The surface area of the equilateral triangular pyramid is the sum of the area of the base and three times the area of the lateral side.
[tex]A_s=A_b+3A_l[/tex]
Suppose the above equation as equation one.
Area of the base of the equilateral tangle can be given as,
[tex]A_b=\dfrac{\sqrt{3} \times10^2}{4} \\ A_b=43.30[/tex]
The area of the lateral triangle is half or the product of the slant height and the side length. Thus,
[tex]A_l=\dfrac{10l}{2} \\ A_l=5l[/tex]
Here l is the slant height of the triangle.
Keep the values in the equation 1,
[tex]\begin{aligned}\\ A_s&=A_b+3A_l\\ 214.5&=43.30+3\times5l\\ \end[/tex]
Solve it for the l,
[tex]\begin{aligned}\\ 15l=214.5-43.3\\ l=\dfrac{171.2}{15} \\ l=11.143\\ \end[/tex]
Thus the slant height of the triangular pyramid is 11.143 centimetres.
Learn more about the slant height of the triangular pyramid here;
https://brainly.com/question/16024003
