Answer:
2. Circumcenter is (-2, 1)
4. Orthocenter is (5, 0)
6. II ∠DOS and ∠CAT are vertical and III ∠DOS and ∠CAT are adjacent
8. m∠PBC = 30°
10. KV, VM, KM
12. DC is greater
14. x = 12
Explanation:
2. Equation of the perpendicular bisector of AB is x = -2 and the equation of the perpendicular bisector of BC is y = 1.
Circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle.
Hence, circumcentre is (-2, 1).
4. Orthocenter is the point of intersection of the altitudes. Here, AB and BC are two altitudes and B is their point of intersection.
Hence, the orthocenter is (5, 0).
6. Adjacent angles are angles separated by a line and vertical angles are angles vertically opposite to each other.
Hence, statements II and III contradict to each other.
8. If P is equidistant from all the three sides, then P is the center of the in-circle and hence it is the incenter of the triangle.
If P is the incentre, then PB is the internal bisector.
Therefore, ∠PBC =
∠ABC = 30°
10. From the triangle sum property, ∠ M = 180 - (110 + 40) = 30.
Since, the side opposite to the smallest is the shortest, the correct order is:
KV, VM, KM
12. Clearly, ∠ ABD is acute. Therefore, ∠ DBC > ∠ ABD.
Since, side opposite to larger angle is greater, DC is greater.
14. Let the sides of the top triangle be p and q. So, two sides of whole triangle are 2p and 2q.
Now, if we compare the top small triangle and the whole triangle, they are similar by SAS similarity since,
the top angle is common and
.
Since, corresponding parts of similar triangles are proportionate,
6x = 5x + 12
6x - 5x = 12
x = 12