Two corners of an isosceles triangle are at #(7 ,9 )# and #(5 ,3 )#. If the triangle's area is #64 #, what are the lengths of the triangle's sides?

Question

Two corners of an isosceles triangle are at #(7   ,9   )# and #(5  ,3    )#.  If the triangle's area is #64   #, what are the lengths of the triangle's sides?

Caroline Aucoin · Accepted Answer

the sides are:#2sqrt10=6.32456#
and #sqrt(10490)/5=20.4841#
and #sqrt(10490)/5=20.4841# To compute for the sides, we need to obtain the height #h# using the area#=64# and the base #b# that can be solved using the points (7, 9) and (5, 3) We solve for the base first #b=sqrt((7-5)^2+(9-3)^2)# #b=sqrt(4+36)=sqrt(40)=2sqrt10# Area#=1/2*b*h# #64=1/2*2*sqrt10*h# #h=64/sqrt10=(32sqrt10)/5# the height h divides the triangle into 2 equal parts and it passes thru the midpoint of the base b. So , we have a right triangle formed Let x and x be the two unknown equal sides #x=sqrt((b/2)^2+h^2)=sqrt(((2sqrt10)/2)^2+((32sqrt10)/5)^2)# #x=sqrt(10+(1024(10))/25)=sqrt((250+10240)/25)# #x=sqrt((10490)/25)# #x=sqrt(10490)/5# #x=20.4841" "# units God bless....I hope the explanation is useful.

Sophia Alfred · Answer

undefined To find the lengths of the sides of the isosceles triangle, we can first calculate the distance between the two given points, which represent the base of the triangle. Then, we can use the formula for the area of a triangle, A = (1/2) * base * height, to find the height of the triangle. Once we have the height, we can use the Pythagorean theorem to find the lengths of the other two sides, which are congruent in an isosceles triangle.

1. Calculate the distance between the two given points (7, 9) and (5, 3) using the distance formula:  
   Distance = √((x2 - x1)^2 + (y2 - y1)^2)

2. Once you have the distance, use the formula for the area of a triangle to find the height:  
   A = (1/2) * base * height

3. Solve for the height using the given area and the base (which is the distance between the two given points).

4. Use the height and one of the sides (which is the distance between one of the given points and the vertex of the triangle) to find the length of the other two sides using the Pythagorean theorem:  
   Side Length = √(height^2 + (1/2 * base)^2)

5. Since the triangle is isosceles, the lengths of the other two sides will be equal. So, once you find the length of one of the other sides, you have the lengths of all three sides of the triangle.