Triangle XYZ below is an isosceles triangle with legs XY and ZY. QR and QS are perpendicular bisectors of triangle XYZ. Find the length of segment QR if
QR = ½ x and QS = -½ x + 4.
what is a perpendicular bisector of a triangle?
To find the length of segment QR, we can use the fact that QR is the perpendicular bisector of triangle XYZ and isosceles triangle XYZ has legs XY and ZY.
Since QR is a perpendicular bisector, it divides XY and ZY into equal segments.
Let's assume the length of XY and ZY to be "a".
Therefore, using the fact that QR divides XY into two equal segments, we have:
XY = QR + QR = 2 * QR
Similarly, QR divides ZY into two equal segments, so we have:
ZY = QR + QR = 2 * QR
Since XY = ZY = a, we can equate these two expressions:
2 * QR = 2 * QR = a
Now, let's solve for QR:
2 * QR = a
Dividing both sides by 2:
QR = a/2
Given the information that QR = ½ x, we can equate this with a/2:
½ x = a/2
Now, let's solve for x:
x = a
Replacing x in the equation QS = -½ x + 4, we have:
QS = -½ a + 4
Therefore, the length of segment QR is a/2 or x/2.
To find the length of segment QR, we need to use the information given and apply some properties of isosceles triangles.
First, let's draw a diagram of triangle XYZ:
X
/ \
/ \
/ \
/ \
/_________\
Y Z
We are told that XY and ZY are the legs of the isosceles triangle. Since it is an isosceles triangle, it means that XY = ZY.
We are also given that QR and QS are perpendicular bisectors of triangle XYZ. Perpendicular bisectors in a triangle are the lines that divide the sides into two equal parts and are perpendicular to the sides. This means that QR and QS bisect the sides XY and XZ, dividing them into equal parts.
Let's calculate the length of segment QR using the given equation QR = ½x:
QR = ½x
Now, let's find the length of the other side QS using the given equation QS = -½x + 4:
QS = -½x + 4
Since QS bisects the side XZ, we can say that QX = XZ/2. Since QX is equal to QS and XZ is equal to XY + YZ, we can substitute these values into the equation:
-½x + 4 = (XY + YZ)/2
Since XY = ZY (as it is an isosceles triangle), we can substitute XY for ZY:
-½x + 4 = (XY + XY)/2
Simplifying the equation:
-½x + 4 = 2XY/2
-½x + 4 = XY
Now, we have two equations:
QR = ½x
-½x + 4 = XY
Since the length of QR is equal to the length of XY, we can equate these two equations:
½x = XY
Substituting this value back into the second equation:
-½x + 4 = (½x)
Simplifying the equation:
4 = x
Therefore, the length of segment QR is equal to ½x, which is equal to ½(4), which results in QR = 2.
Hence, the length of segment QR is 2 units.