Relationships in Triangles: math test solutions, Grade 10 – download the PDF

1 Midsegment / 3 pts

1. \(BC=2\,DE\): \(4x+6=6x-2\), so \(x=4\) (1 pt). \(DE=11\text{ cm}\) and \(BC=22\text{ cm}\) (1 pt).
2. \(AD=15\), \(AE=13\), \(DE=11\), so the perimeter is \(39\text{ cm}\) (1 pt).

2 Centroid / 4 pts

(a) \(BG=2\,GN\): \(3x+4=2x+8\), so \(x=4\) (1 pt). \(BG=16\) and \(BN=16+8=24\) (1 pt).

(b) \(G=\left(\dfrac{-3+5+7}{3},\dfrac{2+6-2}{3}\right)=(3,2)\) (1 pt). The midpoint of \(\overline{QR}\) is \((6,2)\), the median has length \(9\), and \(PG=6=\dfrac{2}{3}\cdot 9\) (1 pt).

3 Triangle inequality / 4 pts

1. (i) \(8+11=19<20\): no. (ii) \(9+12=21>20\): yes. (iii) \(13+13=26>25\): yes. (iv) \(2.5+3.5=6\), not greater: no. (2 pts, 0.5 pt each)
2. \(19-12
3. The largest whole number is 30, so the perimeter is \(12+19+30=61\) (1 pt).

4 Sides and angles / 3 pts

\(10x+20=180\), so \(x=16\) (1 pt). \(\angle P=53^\circ\), \(\angle Q=57^\circ\), \(\angle R=70^\circ\) (1 pt). \(\overline{QR}\) faces \(\angle P\), \(\overline{PR}\) faces \(\angle Q\), \(\overline{PQ}\) faces \(\angle R\), so \(QR

5 Right triangle centers / 3 pts

1. At the midpoint of the hypotenuse (1 pt).
2. The hypotenuse is \(\sqrt{81+144}=15\), so \(R=7.5\text{ in}\) (1 pt).
3. \(K=54\), \(s=18\), so \(r=\dfrac{54}{18}=3\text{ in}\) (1 pt).

6 Orthocenter / 3 pts

The altitude from \(C\) is \(x=4\) (1 pt). The slope of \(\overline{BC}\) is \(\dfrac{6}{4-10}=-1\), so the altitude from \(A\) is \(y=x\) (1 pt). Their intersection is \(H=(4,4)\) (1 pt).