Answer:
a) The length of the mid-segment is 6.25 cm
b) The length of AT = 33 units
c) The value of x is 3
Step-by-step explanation:
a)
* Lets explain the mid-segment of a triangle
- A mid-segment of a triangle is a segment connecting the midpoints
of two sides of a triangle
- This segment has two special properties
# It is parallel to the third side
# The length of the mid-segment is half the length of the third side
∵ The triangle is equilateral triangle
∴ All sides are equal in length
∵ the side lengths = 12.5 cm
∵ The length of the mid-segment = 1/2 the length of the third side
∴ The length of the mid-segment = 1/2 × 12.5 = 6.25 cm
* The length of the mid-segment is 6.25 cm
b)
∵ UT is a perpendicular bisector of AB
∵ T lies on AB
∴ T is the mid-point of AB
∵ AT = BT
∵ AT = 3x + 6
∵ BT = 42 - x
- Equate AT and BT
∴ 3x + 6 = 42 - x
- Add x to both sides
∴ 4x + 6 = 42
- Subtract 6 from both sides
∴ 4x = 36
- Divide both sides by 4
∴ x = 9
∵ AT = 3x + 6
- Substitute x by 9
∴ AT = 3(9) + 6 = 27 + 6 = 33
* The length of AT = 33 units
c)
- In Δ ABC
∵ AB = BC
∴ Δ ABC is an isosceles triangle
∵ BD bisects angle ABC
- In the isosceles Δ the bisector of the vertex angle bisects the base
of the triangle which is opposite to the vertex angle
∵ AC is the opposite side of the vertex B
∴ BD bisects the side AC at D
∴ AD = CD
∵ AD = 5x + 10
∵ CD = 28 - x
∴ 5x + 10 = 28 - x
- Add x to both sides
∴ 6x + 10 = 28
- Subtract 10 from both sides
∴ 6x = 18
- Divide both sides by 6
∴ x = 3
* The value of x is 3
The true statements are:
a) The length of the midsegment is 6.25 cm
b) The length of AT = 33 units
c) The value of x is 3
The length of the midsegment
The length of the triangle is given as
L =12.5cm
So, the length of the midsegment is:
M = 0.5 * L
This gives
M = 0.5 * 12.5 cm
M = 6.25 cm
Hence, the length of the midsegment is 6.25 cm
The length of AT
The given parameters are:
AT = 3x + 6 and TB = 42 - x.
Since point T is the perpendicular bisector, then we have:
3x + 6 = 42 - x
Collect like terms
3x +x = -6 + 42
Evaluate
4x = 36
Divide both sides by 4
x = 19
Recall that:
AT = 3x + 6
So, we have:
At = 3 * 9 + 6
At = 33
Hence. the length of AT = 33 units
The value of x
We have:
AD = 5x + 10
DC = 28 - x
So, we have:
5x + 10 =28 - x
Collect like terms
5x + x = 28 -10
6x =18
Divide
x =3
Hence, the value of x is 3
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