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

1 Find the midsegment ★★★

In \(\triangle ABC\), \(M\) and \(N\) are the midpoints of \(\overline{AB}\) and \(\overline{AC}\).

1. If \(BC=18\text{ cm}\), find \(MN\).
2. If \(MN=7.5\text{ in}\), find \(BC\).

2 Equal distances ★★★

Point \(P\) lies on the perpendicular bisector of \(\overline{AB}\). We know \(PA=3x+4\) and \(PB=5x-6\) (in feet). Find \(x\) and \(PA\).

3 Splitting a median ★★★

\(G\) is the centroid of \(\triangle ABC\) and \(M\) is the midpoint of \(\overline{BC}\).

1. If \(AM=15\text{ cm}\), find \(AG\) and \(GM\).
2. If \(GM=4\text{ in}\), find \(AM\).

4 Can it be a triangle? ★★★

Decide whether each set of lengths can be the sides of a triangle. Justify.

1. 5, 8, 14
2. 6, 9, 14
3. 7, 7, 14
4. 10.5, 4.5, 6.2

5 Order the angles ★★★

In \(\triangle ABC\), \(AB=7\text{ in}\), \(BC=12\text{ in}\) and \(CA=9\text{ in}\). List the angles from smallest to largest.

6 A right triangle’s circumcenter ★★★

A right triangle has legs 6 cm and 8 cm. Where is its circumcenter, and what is the radius of its circumscribed circle?

7 Name the center ★★★

Name the center described in each case.

1. A point equidistant from the three vertices.
2. A point equidistant from the three sides.
3. The balance point of a cardboard triangle.
4. The meeting point of the three heights.
5. Which two of these four centers are always inside the triangle?

8 Midsegment with algebra ★★★

\(DE\) is a midsegment of \(\triangle ABC\) with \(D\) on \(\overline{AB}\), \(E\) on \(\overline{AC}\), and \(\overline{DE}\parallel\overline{BC}\). If \(DE=4x-3\) and \(BC=6x+10\), find \(x\), \(DE\) and \(BC\).

9 Midsegment on a grid ★★★

Triangle \(ABC\) has \(A(-2,1)\), \(B(6,5)\), \(C(2,-5)\). Let \(D\) and \(E\) be the midpoints of \(\overline{AC}\) and \(\overline{BC}\).

1. Find \(D\) and \(E\).
2. Show that \(\overline{DE}\parallel\overline{AB}\).
3. Show that \(DE=\dfrac{1}{2}AB\).

10 Centroid of a sail ★★★

A triangular sail has corners at \(A(1,2)\), \(B(7,4)\), \(C(4,9)\) (units are feet on a grid). Find the centroid \(G\), then check that \(G\) divides the median from \(A\) in a 2 : 1 ratio.

11 Inradius of a 5-12-13 triangle ★★★

A right triangle has sides 5 in, 12 in and 13 in. Find the radius of its inscribed circle and the radius of its circumscribed circle.

12 Range of the third side ★★★

Two sides of a triangle measure 9 cm and 14 cm.

1. Write the inequality for the third side \(x\).
2. How many whole-number lengths are possible for \(x\)?

13 Angle bisector distances ★★★

Point \(P\) lies on the bisector of \(\angle XYZ\). Its distance to \(\overrightarrow{YX}\) is \(2x+1\) inches and its distance to \(\overrightarrow{YZ}\) is \(4x-9\) inches. Find \(x\) and the distances.

14 Angles to sides ★★★

In \(\triangle ABC\), \(\angle A=(2x+10)^\circ\), \(\angle B=(3x-5)^\circ\) and \(\angle C=(x+25)^\circ\). Find \(x\), then order the sides \(AB\), \(BC\), \(AC\) from shortest to longest.

15 A perpendicular bisector equation ★★★

Let \(A(-1,2)\) and \(B(5,6)\).

1. Find the equation of the perpendicular bisector of \(\overline{AB}\).
2. Show that \((0,7)\) is on it by comparing its distances to \(A\) and \(B\).

16 Circumcenter on the grid ★★★

Triangle \(ABC\) has \(A(0,0)\), \(B(8,0)\), \(C(2,6)\). Find the circumcenter \(O\) and the circumradius.

17 Orthocenter on the grid ★★★

Use the same triangle \(A(0,0)\), \(B(8,0)\), \(C(2,6)\). Find the orthocenter \(H\) by writing two altitude equations, then check with the third.

18 The Euler line ★★★

For \(A(0,0)\), \(B(8,0)\), \(C(2,6)\), you know the circumcenter \(O=(4,2)\) and the orthocenter \(H=(2,2)\).

1. Find the centroid \(G\).
2. Show that \(H\), \(G\), \(O\) are collinear.
3. Compare the distances \(HG\) and \(GO\).

19 Midpoints of a quadrilateral ★★★

In quadrilateral \(ABCD\), \(P\), \(Q\), \(R\), \(S\) are the midpoints of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{DA}\). The diagonals measure \(AC=12\text{ cm}\) and \(BD=16\text{ cm}\).

1. Prove that \(PQRS\) is a parallelogram.
2. Find the perimeter of \(PQRS\).

20 Where to put the cell tower ★★★

Three towns lie at \(P(0,0)\), \(Q(10,0)\) and \(R(0,24)\), with units in miles. A cell tower must be the same distance from all three towns.

1. Which triangle center gives the location?
2. Find its coordinates and its distance to each town.

21 Trail lengths ★★★

Three lakes are joined by straight trails. Two trails measure 4.2 km and 6.5 km.

1. Between which values must the third trail’s length lie?
2. If the third length is a whole number of kilometers, what are the shortest and longest possibilities?

22 Triangle inequality with algebra ★★★

A triangle has side lengths \(2x\), \(x+5\) and \(3x-4\). Find all values of \(x\) for which such a triangle exists.