Points E, D, and H are the midpoints of the sides of TUV. UV=82, TV,=102 and HD. Find TU=82.
To begin, we can draw a diagram of triangle TUV and label the given information:
UV = 82
TV = 102
We are trying to find TU.
Next, since E, D, and H are midpoints of the sides of triangle TUV, we can use the midpoint formula to find the lengths of the segments TE, TD, and TH.
The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are:
Midpoint = ( (x1 + x2)/2 , (y1 + y2)/2 )
Using this formula, we can find the lengths of the segments TE, TD, and TH:
TE: Since E is the midpoint of TU, we can find its coordinates as follows:
x-coordinate of E = (x-coordinate of T + x-coordinate of U)/2 = (0 + 82)/2 = 82/2 = 41
y-coordinate of E = (y-coordinate of T + y-coordinate of U)/2 = (0 + 0)/2 = 0/2 = 0
Therefore, E is located at the point (41, 0).
Using the distance formula to find the length of TE:
TE = √((x-coordinate of T - x-coordinate of E)^2 + (y-coordinate of T - y-coordinate of E)^2)
= √((0 - 41)^2 + (0 - 0)^2)
= √((-41)^2 + (0)^2)
= √(1681 + 0)
= √1681
= 41
TD: Since D is the midpoint of UV, we can find its coordinates as follows:
x-coordinate of D = (x-coordinate of U + x-coordinate of V)/2 = (0 + 0)/2 = 0/2 = 0
y-coordinate of D = (y-coordinate of U + y-coordinate of V)/2 = (0 + 0)/2 = 0/2 = 0
Therefore, D is located at the point (0, 0).
Using the distance formula to find the length of TD:
TD = √((x-coordinate of T - x-coordinate of D)^2 + (y-coordinate of T - y-coordinate of D)^2)
= √((0 - 0)^2 + (0 - 0)^2)
= √((0)^2 + (0)^2)
= √(0 + 0)
= √0
= 0
TH: Since H is the midpoint of TV, we can find its coordinates as follows:
x-coordinate of H = (x-coordinate of T + x-coordinate of V)/2 = (0 + 0)/2 = 0/2 = 0
y-coordinate of H = (y-coordinate of T + y-coordinate of V)/2 = (0 + 102)/2 = 102/2 = 51
Therefore, H is located at the point (0, 51).
Using the distance formula to find the length of TH:
TH = √((x-coordinate of T - x-coordinate of H)^2 + (y-coordinate of T - y-coordinate of H)^2)
= √((0 - 0)^2 + (0 - 51)^2)
= √((0)^2 + (-51)^2)
= √(0 + 2601)
= √2601
= 51
Now, we can use the fact that E, D, and H are midpoints of the sides of triangle TUV to find the lengths of TU, TV, and UV:
TU = 2*TE = 2*41 = 82
TV = 2*TH = 2*51 = 102
UV = 2*TD = 2*0 = 0
Therefore, we have found that TU = 82.