1. The perimeter of an isosceles triangle is 50cm and the length of the altitude to the base is 5cm. Find the length of the base.
2. In quadrilateral QRST:
TQ=8cm; (angle)R=60; (angle)T=90; QR=RS; ST=8cm.
Find the exact length of the longer diagonal of the quadrilateral.
If the equal sides are s and the base is b,
2s+b = 50
(b/2)^2 + 5^2 = s^2
For QRST, TQS is an isosceles right triangle, so QS = 8√2
QRS is isosceles, with vertex angle 60°, so QR=RS=√8 = 2√2. So, QRS is equilateral.
So diagonal TR is the sum of the two altitudes of the isosceles triangles:
4√6+4√2 = 4(√6+√2)
To find the length of the base of an isosceles triangle:
Step 1: Recall that an isosceles triangle has two equal sides.
Step 2: Let's assume that the two equal sides are each of length x cm.
Step 3: Since the perimeter of the triangle is 50 cm, the sum of all three sides is 50 cm. Therefore, we have:
x + x + b = 50,
where b represents the length of the base.
Step 4: Simplify the equation:
2x + b = 50.
Step 5: We also know that the length of the altitude to the base is 5 cm. Recall that for an isosceles triangle, the altitude bisects the base. Therefore, we have:
b/2 = 5.
Step 6: Solve the equation for b by substituting the value of b from Step 5 into the equation from Step 4:
2x + (2 * 5) = 50,
2x + 10 = 50,
2x = 40,
x = 20.
Step 7: Substitute the value of x into the equation b/2 = 5 to find the length of the base:
b/2 = 5,
b = 10.
Therefore, the length of the base of the isosceles triangle is 10 cm.
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To find the exact length of the longer diagonal of the quadrilateral:
Step 1: Start by drawing QRST based on the given information.
Step 2: Since (angle) T = 90, QRST is a rectangle.
Step 3: Since QR = RS, QRST is also a rhombus (a special type of rectangle where all sides are equal).
Step 4: Since the diagonals of a rectangle bisect each other, we can cut the rhombus QRST into two congruent right-angled triangles: QRT and STR.
Step 5: In triangle QRT, QRT is a 30-60-90 triangle where (angle) R = 60 and along with (angle) T = 90.
Step 6: Using trigonometry, we know that in a 30-60-90 triangle, the length of the shorter leg is half the length of the hypotenuse. Therefore, QT = TQ/2 = 4 cm.
Step 7: Since QR = RS, SQ = ST - TQ = 8 - 4 = 4 cm.
Step 8: Now we can calculate the length of the longer diagonal by using the Pythagorean theorem in triangle STR:
SR^2 = ST^2 + TR^2,
SR^2 = (8)^2 + (4)^2,
SR^2 = 64 + 16,
SR^2 = 80.
Step 9: Take the square root of both sides of the equation:
SR = √80.
Step 10: Simplify the square root:
SR = 4√5.
Therefore, the exact length of the longer diagonal of quadrilateral QRST is 4√5 cm.