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

1 Midsegment / 3 pts

In \(\triangle ABC\), \(D\) and \(E\) are the midpoints of \(\overline{AB}\) and \(\overline{AC}\). We have \(DE=3x-1\) and \(BC=4x+6\) (in centimeters).

1. Find \(x\), \(DE\) and \(BC\).
2. If \(AB=30\text{ cm}\) and \(AC=26\text{ cm}\), find the perimeter of \(\triangle ADE\).

2 Centroid / 4 pts

(a) \(G\) is the centroid of \(\triangle ABC\) and \(N\) is the midpoint of \(\overline{AC}\). If \(BG=3x+4\) and \(GN=x+4\), find \(x\), \(BG\) and \(BN\).

(b) Triangle \(PQR\) has \(P(-3,2)\), \(Q(5,6)\), \(R(7,-2)\). Find its centroid, then verify that it lies \(\dfrac{2}{3}\) of the way along the median from \(P\).

3 Triangle inequality / 4 pts

1. Which of these lengths can form a triangle? (i) 8, 11, 20 (ii) 9, 12, 20 (iii) 13, 13, 25 (iv) 2.5, 3.5, 6
2. Two sides measure 12 and 19. Write the range of the third side \(x\).
3. If the third side is a whole number, what is the largest possible perimeter?

4 Sides and angles / 3 pts

In \(\triangle PQR\), \(\angle P=(3x+5)^\circ\), \(\angle Q=(2x+25)^\circ\), \(\angle R=(5x-10)^\circ\). Find \(x\), the three angles, and order the sides \(PQ\), \(QR\), \(PR\) from shortest to longest.

5 Right triangle centers / 3 pts

A right triangle has legs 9 in and 12 in.

1. Where is its circumcenter?
2. Find the circumradius.
3. Find the inradius.

6 Orthocenter / 3 pts

Triangle \(ABC\) has \(A(0,0)\), \(B(10,0)\), \(C(4,6)\). Find the equation of the altitude from \(C\), the equation of the altitude from \(A\), and the orthocenter.