Question

In a triangle \(ABC\), \(AB=3\), \(BC=4\) and \(CA=5\). Point \(D\) is the midpoint of \(AB\), point \(E\) is on segment \(AC\) and point \(F\) is on segment \(BC\). If \(AE=1.5\) and \(BF=0.5\), then \(\angle DEF =\)

• A. 30 degrees
• B. 45 degrees
• C. 60 degrees
• D. 75 degrees

Hint:

Spot the two isosceles triangles formed at A and at C using the given lengths, before trying to chase angles at E.

Answer 1

The Correct Option is B

Step 1: Set up coordinates using the right angle.
Since $AB=3$, $BC=4$, $CA=5$ satisfy $3^2+4^2=5^2$, the right angle sits at $B$. Place $B=(0,0)$, $A=(0,3)$ along the y-axis (so $AB=3$), and $C=(4,0)$ along the x-axis (so $BC=4$). Check: $CA=\sqrt{(4-0)^2+(0-3)^2}=\sqrt{16+9}=5$, correct.

Step 2: Find the coordinates of D, E, F.
$D$ is the midpoint of $AB$: $D=(0,1.5)$. $E$ is on $AC$ with $AE=1.5$; since $AC=5$, moving from $A=(0,3)$ toward $C=(4,0)$ along the unit direction $\left(\frac{4}{5},-\frac{3}{5}\right)$ gives $$E = \left(0+1.5\times\tfrac{4}{5},\ 3+1.5\times(-\tfrac{3}{5})\right) = (1.2,\ 2.1)$$ $F$ is on $BC$ with $BF=0.5$, so moving from $B=(0,0)$ toward $C=(4,0)$ along the x-axis, $F=(0.5,0)$.

Step 3: Form the direction vectors from E. $$\vec{ED} = D - E = (-1.2,\ -0.6)$$ $$\vec{EF} = F - E = (-0.7,\ -2.1)$$

Step 4: Use the dot product to find the angle between them. $$\vec{ED}\cdot\vec{EF} = (-1.2)(-0.7)+(-0.6)(-2.1) = 0.84+1.26 = 2.10$$ $$|\vec{ED}| = \sqrt{1.44+0.36} \approx 1.342, \quad |\vec{EF}| = \sqrt{0.49+4.41} \approx 2.214$$ $$\cos(\angle DEF) = \frac{2.10}{1.342\times2.214} \approx \frac{2.10}{2.970} \approx 0.7071$$

Step 5: Identify the angle.
$\cos^{-1}(0.7071) = 45^{\circ}$, since $\cos45^{\circ}=\frac{1}{\sqrt2}\approx0.7071$.

Final Answer:
$\angle DEF = 45^{\circ}$, confirming the same result by a completely independent, coordinate based method. $$\boxed{45^{\circ}}$$