Question

points d and e are midpoints of the sides of triangle abc. the perimeter of the triangle is 48 units. what is the value of t? 2 3 6 8

Explanation:

Step1: Use the property of mid - points

Since \(D\) and \(E\) are mid - points of the sides of \(\triangle ABC\), the perimeter of \(\triangle ABC\) is \(2\times(3t + 4t+7t + 6)\) (by the mid - point theorem, each side of the small segments is half of the corresponding side of the large triangle).
The perimeter of \(\triangle ABC\) is given as \(48\) units. So, \(3t+4t + 7t+6=24\) (because the sum of the lengths of the segments formed by mid - points is half of the perimeter of the whole triangle).

Step2: Simplify the equation

Combine like terms: \((3 + 4+7)t+6 = 24\), which is \(14t+6 = 24\).
Subtract \(6\) from both sides: \(14t=24 - 6=18\).

Step3: Solve for \(t\)

\(14t = 18\) is incorrect. Wait, no, actually, since \(D\) is the mid - point of \(AB\), \(AB = 2\times3t=6t\), \(E\) is the mid - point of \(BC\), \(BC = 2\times4t = 8t\), \(AC=7t + 6\).
The perimeter \(P=AB + BC+AC\), so \(6t+8t+7t + 6=48\).
Combine like terms: \((6 + 8+7)t+6=48\), \(21t+6 = 48\).
Subtract \(6\) from both sides: \(21t=48 - 6=42\).
Divide both sides by \(21\): \(t=\frac{42}{21}=2\).

Answer:

A. 2